Problem:
Real numbers a,b, and c have arithmetic mean 0 . The arithmetic mean of a2,b2, and c2 is 10 . What is the arithmetic mean of ab,ac, and bc ?
Answer Choices:
A. −5
B. −310
C. −910
D. 0
E. 910
Solution:
The given information implies that a+b+c=0 and a2+b2+c2=30. Then
0=(a+b+c)2=a2+b2+c2+2ab+2ac+2bc=30+2(ab+ac+bc)
Therefore 2(ab+ac+bc)=−30 and the requested arithmetic mean is 3ab+ac+bc=3−15=(A)−5.
OR
Consider the system of equations implied by the conditions of the problem,
a+b+ca2+b2+c2=0=30
and suppose that a=0. Then b+c=0, so b=−c, and substituting into the second equation gives 2b2=30, from which b=±15 and c=∓15. If one assumes that the requested arithmetic mean is determined by the given information, independent of the value of a, then